Weighted uniform Diophantine approximation of systems of linear forms

TL;DR

This paper explores uniform Diophantine approximation of linear forms with weighted norms, establishing Dirichlet's theorem in this context and showing that matrices that can be approximated better than Dirichlet's bound are prevalent in terms of Hausdorff dimension.

Contribution

It introduces weighted uniform approximation properties for matrices using general norms and connects these properties to homogeneous dynamics, extending classical results to new norm settings.

Findings

Proved a precise form of Dirichlet's theorem for certain norms.

Showed the set of Dirichlet-improvable matrices has full Hausdorff dimension.

Established the connection between approximation properties and trajectories in homogeneous spaces.

Abstract

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({\mathbb Z})$ avoiding a compact subset of the space of lattices called the critical locus defined with respect to the corresponding norm. The trajectory is formed by the action of a one-parameter diagonal subgroup corresponding to the weights. We first state a very precise form of Dirichlet's theorem and prove it for some norms. Secondly we show, for these same norms, that the set of Dirichlet-improvable matrices has full Hausdorff dimension. Though the techniques used vary greatly depending on the chosen norm, we expect these results to hold in general.